*The nature of the misconception of the halting problem*

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qy ∞
if Ĥ applied to ⟨Ĥ⟩ halts, and

Ĥ.q0 ⟨Ĥ⟩ ⊢* Ĥ.qx ⟨Ĥ⟩ ⟨Ĥ⟩ ⊢* Ĥ.qn
>>>if Ĥ applied to ⟨Ĥ⟩ does not halt<<<
(Linz:1990:320) adapted with clearer notational conventions.

[Linz 315-320](https://www.liarparadox.org/Peter_Linz_HP_315-320.pdf)

We begin with the premise that the Linz H is a simulating halt decider.

H has been embedded in Ĥ at state Ĥ.qx. The H.qy path has been disabled
in Ĥ by appending an infinite loop.

The H.qn path of the original H that is embedded at Ĥ.qx is be named
[HALF OF COMPUTABLE FUNCTION H]:HOCFH

Now we get to the misconception about the halting problem:
Linz and everyone else believes that the condition of this Ĥ.qn path is
>>>if Ĥ applied to ⟨Ĥ⟩ does not halt<<< as if Ĥ was reporting on whether
or not itself halts. This is incorrect.

HOCFH does not transition to Ĥ.qn: >>>if Ĥ applied to <Ĥ> does not halt<<<

HOCFH transitions to Ĥ.qn when its simulation of <Ĥ> applied to <Ĥ>
never reaches the final state of <Ĥ> applied to <Ĥ>.

The input to HOCFH is ⟨Ĥ⟩ ⟨Ĥ⟩.
The input to HOCFH is not Ĥ ⟨Ĥ⟩.

HOCFH does not compute whether or not >>>Ĥ applied to <Ĥ> does not
halt<<< because an actual Turing machine cannot be input to any
computable function.

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Copyright 2021 Pete Olcott

Talent hits a target no one else can hit;
Genius hits a target no one else can see.
Arthur Schopenhauer